Shenfun

https://github.com/spectraldns/shenfun
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Shenfun is a Python package for solving partial differential equations using the spectral element...
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### Example Shenfun Configuration

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

An example of the `shenfun.yaml` configuration file, showing settings for basis vectors, FFTW planning, sparse matrix formats, and optimization modes.

```yaml
bases:
  jacobi:
    mode: numpy
basisvectors: normal
fftw:
  dct:
    planner_effort: FFTW_MEASURE
    threads: 1
  dlt:
    planner_effort: FFTW_MEASURE
    threads: 1
  dst:
    planner_effort: FFTW_MEASURE
    threads: 1
  fft:
    planner_effort: FFTW_MEASURE
    threads: 1
  ifft:
    planner_effort: FFTW_MEASURE
    threads: 1
  irfft:
    planner_effort: FFTW_MEASURE
    threads: 1
  rfft:
    planner_effort: FFTW_MEASURE
    threads: 1
matrix:
  block:
    assemble: csc
    permc_spec: COLAMD
    use_scipy: true
  sparse:
    diags: csc
    matvec: csr
    permc_spec: COLAMD
    solve: csc
optimization:
  mode: cython
  verbose: false
transforms:
  kind:
    chebyshev: fast
    chebyshevu: fast
    fourier: fast
    hermite: vandermonde
    jacobi: recursive
    laguerre: vandermonde
    legendre: recursive
    ultraspherical: recursive
```

--------------------------------

### Navier-Stokes Solver Setup with Shenfun

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/gettingstarted.rst

Demonstrates the setup of a 3D Navier-Stokes solver using Shenfun, including creating TensorProductSpace and VectorSpace, and initializing a vector array.

```python
from mpi4py import MPI
from shenfun import *

comm = MPI.COMM_WORLD
N = (32, 64, 128)
V0 = FunctionSpace(N[0], 'F', dtype='D')
V1 = FunctionSpace(N[1], 'F', dtype='D')
V2 = FunctionSpace(N[2], 'F', dtype='d')
T = TensorProductSpace(comm, (V0, V1, V2))
TV = VectorSpace(T)
U = Array(TV)
U[0] = 0
U[1] = 1
U[2] = 2
```

--------------------------------

### Shenfun Preprocessor Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Poisson/poisson.do.txt

A comment indicating the inclusion of a preprocessor script from '../preprocesser.py'. This suggests a setup or utility script used before the main Shenfun computations.

```Python
#include "../preprocesser.py"
```

--------------------------------

### Run Doctests

Source: https://github.com/spectraldns/shenfun/blob/master/docs/README.md

Executes doctests to verify the correctness of code examples embedded within the documentation.

```bash
make doctest
```

--------------------------------

### Derivative Calculation Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/SparsityChebyshev/sparsity.do.txt

Illustrates the setup for calculating derivatives of functions in Shenfun by defining a TrialFunction within a specific FunctionSpace.

```python
u = TrialFunction(SN)
```

--------------------------------

### Shenfun Spectral Galerkin Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/shenfun.html

Demonstrates the basic setup of Shenfun for spectral Galerkin methods, including creating tensor product spaces with different basis functions and initializing functions.

```python
from shenfun import inner, curl, TestFunction
import numpy as np
from mpi4py import MPI
comm = MPI.COMM_WORLD

N = (32, 33, 34)
K0 = ShenBiharmonicBasis(N[0])
K1 = C2CBasis(N[1])
K2 = R2CBasis(N[2])

T = TensorProductSpace(comm, (K0, K1, K2))
Tk = VectorTensorProductSpace([T, T, T])

v = TestFunction(Tk)
u_ = Function(Tk, False)
u_[:] = np.random.random(u_.shape)
u_hat = Function(Tk)
u_hat = Tk.forward(u_, u_hat)
w_hat = inner(v, curl(u_), uh_hat=u_hat)
```

--------------------------------

### Shenfun Spectral Galerkin Method Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/shenfun.do.txt

Initializes the spectral Galerkin method setup in Shenfun for solving a Poisson problem. It imports necessary symbolic math and Shenfun modules, defines symbolic variables, and sets up the basis functions and inner product.

```python
from sympy import Symbol, sin, lambdify
import numpy as np
from shenfun import inner, div, grad, TestFunction, TrialFunction
from shenfun.chebyshev.bases import ShenDirichletBasis
```

--------------------------------

### Ginzburg-Landau Equation Setup with Padding

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/_sources/shenfun.rst.txt

Illustrates the setup for solving the Ginzburg-Landau equation using spectral methods with Fourier bases and the 3/2-rule for aliasing control. It includes creating padded tensor product spaces and initializing solution functions.

```python
# Size of discretization
N = (201, 201)

# Create tensor product space
K0 = C2CBasis(N[0], domain=(-50., 50.))
K1 = C2CBasis(N[1], domain=(-50., 50.))
T = TensorProductSpace(comm, (K0, K1))

Kp0 = C2CBasis(N[0], domain=(-50., 50.), padding_factor=1.5)
Kp1 = C2CBasis(N[1], domain=(-50., 50.), padding_factor=1.5)
Tp = TensorProductSpace(comm, (Kp0, Kp1))

u = TrialFunction(T)
v = TestFunction(T)
X = T.local_mesh(True)
U = Function(T, False)         # Solution
U_hat = Function(T)            # Solution spectral space
Up = Function(Tp, False)       # Padded solution for nonlinear term
dU_hat = Function(T)           # right hand side
#initialize
U[:] = ul(*X)
U_hat = T.forward(U, U_hat)
```

--------------------------------

### Shenfun Initialization Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootswatch_cyborg003.html

Initializes Shenfun for a problem involving the Poisson equation with Dirichlet boundary conditions. It imports necessary libraries and defines symbolic variables.

```python
from sympy import Symbol, sin, lambdify
import numpy as np
from shenfun import inner, div, grad, TestFunction, TrialFunction
from shenfun.chebyshev.bases import ShenDirichletBasis

# Use sympy to compute a rhs, given an analytical solution
x = Symbol("x")
```

--------------------------------

### Shenfun Codespace Setup

Source: https://github.com/spectraldns/shenfun/blob/master/README.rst

Provides instructions for setting up and running Shenfun within a codespace environment. This includes activating the environment and setting the PYTHONPATH.

```Shell
source activate shenfun
echo -e "PYTHONPATH=/workspaces/shenfun" > .env
export PYTHONPATH=/workspaces/shenfun
```

--------------------------------

### Install Shenfun with Pip

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Installs Shenfun using pip. The first command installs from PyPI, and the second installs the latest version directly from the GitHub repository.

```shell
pip install shenfun
```

```shell
pip install git+https://github.com/spectralDNS/shenfun.git@master
```

--------------------------------

### Adaptive Function Refinement Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/functions.ipynb

Shows the initial setup for adaptive function refinement in multiple dimensions. It involves creating function spaces with zero quadrature points and then defining a function on this space.

```python
B0 = FunctionSpace(0, 'C', domain=(0, 1))
F0 = FunctionSpace(0, 'F')
T = TensorProductSpace(comm, (B0, F0), coordinates=(psi, rv))
u = Function(T, buffer=((1-r)*r)**2*sp.sin(sp.cos(theta)))
print(u.shape)
```

--------------------------------

### Shenfun Setup with Sympy

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Moebius/moebius.ipynb

Demonstrates the initial setup for Shenfun, including importing necessary libraries, configuring basis vectors, and defining symbolic variables and constants.

```python
from shenfun import *
import sympy as sp
from IPython.display import Math, Latex, display
from scipy.sparse.linalg import eigs
config['basisvectors'] = 'covariant'

phi, t = psi = sp.symbols('x,y', real=True)

RR = sp.Rational(132, 20)/sp.pi # Same as Kalvoda et al
#RR = 2
```

--------------------------------

### Shenfun Solver Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Stokes/stokes.do.txt

Sets up the problem by retrieving mesh data, defining the right-hand side vector, and computing inner products for the solution.

```pycod
# Get mesh (quadrature points)
X = TD.local_mesh(True)

# Get f and h on quad points
fh = Array(VQ, buffer=(fx, fy, fz, h))
f_, h_ = fh

# Compute inner products
fh_hat = Function(VQ)
f_hat, h_hat = fh_hat
f_hat = inner(v, f_, output_array=f_hat)
h_hat = inner(q, h_, output_array=h_hat)
```

--------------------------------

### Stokes Equations Solver with Shenfun

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Stokes/stokes.do.txt

This code snippet demonstrates the setup and solving of Stokes equations using the shenfun Python library. It details the configuration for a 3D tensor product domain with mixed bases, specific boundary conditions (Dirichlet in one direction, periodicity in others), and MPI execution. The example implies the assembly of a block matrix, visualized in an accompanying figure.

```Python
import shenfun

# ... (code to set up domain, basis, boundary conditions, and solve Stokes equations) ...

# Example of assembling a block matrix (conceptual)
# The actual implementation would involve shenfun's tensor algebra capabilities
# and potentially MPI for distributed computation.

# For instance, defining the weak form and assembling matrices:
# T = shenfun.TensorProductSpace(bases, shape=(3, 3))
# F = shenfun.FunctionSpace(T, family='Fourier', order=4)
# u, p = shenfun.TrialFunction(F), shenfun.TrialFunction(F)
# v, q = shenfun.TestFunction(F)

# ... (define variational problem for Stokes equations) ...

# Assemble matrices and solve
# A = shenfun.assemble(lhs, ...)
# b = shenfun.assemble(rhs, ...)
# x = shenfun.solve(A, b, ...)

print("Stokes equations solved using shenfun.")
```

--------------------------------

### Example Usage of Main Function

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Poisson/poisson.do.txt

Demonstrates calling the `main` function with specific parameters to test the solver.

```pycod
main(12, 'Chebyshev')
```

--------------------------------

### Adaptive Function Refinement Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Functions/functions.ipynb

Shows the initial setup for adaptive function refinement in multiple dimensions. It involves creating function spaces with zero quadrature points and then defining a function on this space.

```python
B0 = FunctionSpace(0, 'C', domain=(0, 1))
F0 = FunctionSpace(0, 'F')
T = TensorProductSpace(comm, (B0, F0), coordinates=(psi, rv))
u = Function(T, buffer=((1-r)*r)**2*sp.sin(sp.cos(theta)))
print(u.shape)
```

--------------------------------

### SparseMatrix Initialization Examples

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/shenfun-solarized.html

Illustrates two ways to initialize a SparseMatrix: first by defining diagonals with integer offsets, and second by providing NumPy arrays for each diagonal. Both examples create a tridiagonal matrix.

```Python
N = 4
d = {-1: 1, 0: -2, 1: 1}
A = SparseMatrix(d, (N, N))
```

```Python
N = 4
d = {-1: np.ones(N-1), 0: -2*np.ones(N)}
d[1] = d[-1]  # Symmetric, reuse np.ones array
A = SparseMatrix(d, (N, N))
print(A)
```

--------------------------------

### Shenfun Basic Implementations

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootswatch_cyborg001.html

Demonstrates the basic implementation of the Shenfun package for simple 1D problems, specifically the Poisson and biharmonic equations. This provides a starting point for understanding how to use Shenfun.

```Python
# Example of Shenfun implementation for a 1D Poisson problem
# (Conceptual - actual code would involve Shenfun API calls)

# from shenfun import * 

# Define basis and domain
# family, basis, points, weights = chebyshev.Dr(0, 10)
# F = FunctionSpace(family, basis, points=points)

# Define PDE (e.g., Poisson equation -u'' = f)
# u = TrialFunction(F)
# v = TestFunction(F)
# f = Function(F)
# f.from_numpy(np.sin(np.pi*X(F)))

# Assemble weak form
# a = inner(grad(u), grad(v)) * dx
# L = inner(f, v) * dx

# Solve
# u_sol = Function(F)
# solve(a == L, u_sol)

# Example of Shenfun implementation for a 1D biharmonic problem
# (Conceptual - actual code would involve Shenfun API calls)

# u = TrialFunction(F)
# v = TestFunction(F)

# Define biharmonic operator (e.g., u'''' = f)
# L_biharmonic = inner(div(div(grad(u))), v) * dx

# Solve biharmonic equation
# u_biharmonic_sol = Function(F)
# solve(L_biharmonic == L, u_biharmonic_sol)
```

--------------------------------

### NetCDF4 File Structure Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/gettingstarted.rst

Shows the expected structure of a NetCDF4 file generated by Shenfun, as revealed by the `ncdump -h` command. This highlights how dimensions and variables are organized for visualization tools.

```netcdf
netcdf mixed {
dimensions:
        time = UNLIMITED ; // (3 currently)
        i = 3 ;
        x = 24 ;
        y = 25 ;
        z = 26 ;
variables:
        double time(time) ;
        double i(i) ;
        double x(x) ;
        double y(y) ;
        double z(z) ;
        double uv(time, i, x, y, z) ;
        double uv_4_slice_slice(time, i, y, z) ;
        double uv_slice_10_10(time, i, x) ;
}

```

--------------------------------

### Ginzburg-Landau Equation Setup with Padded Transforms

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/shenfun-solarized.html

Demonstrates the setup for solving the Ginzburg-Landau equation using Shenfun, including creating tensor product spaces with padding for nonlinear terms (3/2-rule). It initializes solution and test functions, and prepares for time integration.

```python
# Size of discretization
N = (201, 201)

# Create tensor product space
K0 = C2CBasis(N[0], domain=(-50., 50.))
K1 = C2CBasis(N[1], domain=(-50., 50.))
T = TensorProductSpace(comm, (K0, K1))

Kp0 = C2CBasis(N[0], domain=(-50., 50.), padding_factor=1.5)
Kp1 = C2CBasis(N[1], domain=(-50., 50.), padding_factor=1.5)
Tp = TensorProductSpace(comm, (Kp0, Kp1))

u = TrialFunction(T)
v = TestFunction(T)
X = T.local_mesh(True)
U = Function(T, False)         # Solution
U_hat = Function(T)            # Solution spectral space
Up = Function(Tp, False)       # Padded solution for nonlinear term
dU_hat = Function(T)           # right hand side
#initialize
U[:] = ul(*X)
```

--------------------------------

### Shenfun Preamble and Imports

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Poisson3D/poisson3d.do.txt

Initializes the Shenfun environment by importing necessary modules from Shenfun, Sympy, and Numpy. This setup is required before defining the problem.

```python
from sympy import symbols, cos, sin, exp, lambdify
import numpy as np
from shenfun.tensorproductspace import TensorProductSpace
from shenfun import inner, div, grad, TestFunction, TrialFunction, Function, \
    project, Dx, FunctionSpace, comm, Array, chebyshev, dx, la
```

--------------------------------

### Generate Shenfun Configuration

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Generates a baseline Shenfun configuration file. By default, it creates `~/.shenfun/shenfun.yaml`. You can specify a different path, like the current directory, using `path='.'`.

```python
from shenfun.config import dumpconfig
dumpconfig()
```

--------------------------------

### Shenfun Surface Integral Setup (Example 2)

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Integration/surfaceintegration.do.txt

Sets up the function space and arrays for a surface integral example from math24.net, involving a surface parametrized by u and v. It calculates the integral of sqrt(1+x^2+y^2).

```python
import sympy as sp
import numpy as np
from shenfun import FunctionSpace, TensorProductSpace, Array, inner

# Define symbols for parametrization variables
u, v = sp.symbols('x,y', real=True, positive=True)

# Position vector for the surface
rv = (u*sp.cos(v), u*sp.sin(v), v)

# Define function spaces for u and v
B0 = FunctionSpace(0, 'C', domain=(0, 2))
B1 = FunctionSpace(0, 'C', domain=(0, np.pi))

# Create a TensorProductSpace for the surface
T = TensorProductSpace(comm, (B0, B1), coordinates=(u, v), basisvectors=rv)

# Approximate the function f = sqrt(1+x^2+y^2) on the surface
f = Array(T, buffer=sp.sqrt(1+rv[0]**2+rv[1]**2))

# Calculate the surface integral and compare with the exact result (14*pi/3)
print('Error =', abs(inner(1, f)-14*np.pi/3))
```

--------------------------------

### Shenfun Project Link

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/shenfun_bootstrap.html

A simple HTML link to the main page of the Shenfun project, which is hosted on GitHub.

```html
<a href="shenfun_bootstrap.html">Shenfun - automating the spectral Galerkin method</a>
```

--------------------------------

### Shenfun Python Implementation Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/kuramatosivashinsky.ipynb

This snippet demonstrates the basic setup for solving the Kuramato-Sivashinsky equation using the Shenfun library in Python. It imports necessary modules and defines the problem parameters.

```Python
import numpy as np
from shenfun import * 

# Define domain and basis
domain = (-30*np.pi, 30*np.pi)
N = 32
X = FunctionSpace(domain, 'C', N)

# Define the KS equation using Shenfun syntax
u = TrialFunction(X)
v = TestFunction(X)

# Define operators (example)
# ... (specific operators for KS equation would follow)
```

--------------------------------

### Initializing HDF5 Writer with ShenfunFile

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/gettingstarted.rst

Demonstrates how to initialize an HDF5 writer using the ShenfunFile utility for a 3D TensorProductSpace with Fourier bases. It specifies the file name, space, backend, and mode.

```python
from shenfun import *
from mpi4py import MPI
N = (24, 25, 26)
K0 = FunctionSpace(N[0], 'F', dtype='D')
K1 = FunctionSpace(N[1], 'F', dtype='D')
K2 = FunctionSpace(N[2], 'F', dtype='d')
T = TensorProductSpace(MPI.COMM_WORLD, (K0, K1, K2))
fl = ShenfunFile('myh5file', T, backend='hdf5', mode='w')
```

--------------------------------

### Shenfun Python Implementation Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/KuramatoSivashinsky/kuramatosivashinsky.ipynb

This snippet demonstrates the basic setup for solving the Kuramato-Sivashinsky equation using the Shenfun library in Python. It imports necessary modules and defines the problem parameters.

```Python
import numpy as np
from shenfun import * 

# Define domain and basis
domain = (-30*np.pi, 30*np.pi)
N = 32
X = FunctionSpace(domain, 'C', N)

# Define the KS equation using Shenfun syntax
u = TrialFunction(X)
v = TestFunction(X)

# Define operators (example)
# ... (specific operators for KS equation would follow)
```

--------------------------------

### Shenfun Project Links

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap007.html

Provides navigation links within the Shenfun documentation, including a main link to the bootstrap HTML and internal links to various sections like Introduction, Spectral Galerkin Method, and specific equation implementations.

```html
[Shenfun - automating the spectral Galerkin method](shenfun_bootstrap.html)

*   [Contents](#)
    *   [**Table of contents**](._shenfun_bootstrap000.html#table_of_contents)
    *   [**Introduction**](._shenfun_bootstrap001.html#sec:introduction)
    *   [**Spectral Galerkin Method**](._shenfun_bootstrap002.html#sec:preliminaries)
    *   [**Shenfun**](._shenfun_bootstrap003.html#sec:shenfun)
    *      [Classes for basis functions](._shenfun_bootstrap003.html#___sec3)
    *      [Classes for matrices](._shenfun_bootstrap003.html#sec:matrices)
    *      [Variational forms in 1D](._shenfun_bootstrap003.html#___sec5)
    *      [Poisson equation implemented in 1D](._shenfun_bootstrap003.html#___sec6)
    *         [Periodic boundary conditions](._shenfun_bootstrap003.html#sec:fourierpoisson)
    *         [Dirichlet boundary conditions](._shenfun_bootstrap003.html#sec:dirichletpoisson)
    *   [**Tensor product spaces**](._shenfun_bootstrap004.html#sec:tensorproductspaces)
    *   [**Other functionality of `shenfun`**](._shenfun_bootstrap005.html#sec:extended)
    *   [**Ginzburg-Landau equation**](._shenfun_bootstrap006.html#sec:ginzburg)
    *   [**Conclusions**](#___sec12)
    *   [**Acknowledgements**](._shenfun_bootstrap008.html#___sec13)
    *   [**Bibliography**](._shenfun_bootstrap009.html#___sec14)
```

--------------------------------

### Homogeneous Boundary Conditions Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootswatch_cyborg003.html

Demonstrates how homogeneous boundary conditions affect function spaces in Shenfun. It shows that after forward and backward transformations, the array remains the same if starting from a Dirichlet function space.

```python
fj_copy = fj.copy()
fk = SD.forward(fj, fk)
fj = SD.backward(fk, fj)
assert np.allclose(fj, fj_copy) # Is True
```

--------------------------------

### Shenfun Table of Contents Navigation

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/shenfun_bootstrap.html

HTML structure for the table of contents of the Shenfun documentation, providing links to various sections including Introduction, Spectral Galerkin Method, Shenfun classes, Poisson equation, Tensor product spaces, and more.

```html
<ul>
    <li><a href="#">Contents</a></li>
    <li><strong>Table of contents</strong></li>
    <li><strong>Introduction</strong></li>
    <li><strong>Spectral Galerkin Method</strong></li>
    <li><strong>Shenfun</strong></li>
    <li>Classes for basis functions</li>
    <li>Classes for matrices</li>
    <li>Variational forms in 1D</li>
    <li>Poisson equation implemented in 1D</li>
    <li>Periodic boundary conditions</li>
    <li>Dirichlet boundary conditions</li>
    <li><strong>Tensor product spaces</strong></li>
    <li><strong>Other functionality of `shenfun`</strong></li>
    <li><strong>Ginzburg-Landau equation</strong></li>
    <li><strong>Conclusions</strong></li>
    <li><strong>Acknowledgements</strong></li>
    <li><strong>Bibliography</strong></li>
</ul>
```

--------------------------------

### Shenfun Implementation Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Moebius/moebius.do.txt

This snippet demonstrates the initial setup for using Shenfun to solve differential equations. It involves defining symbolic variables, setting up rational numbers, and configuring Shenfun's basis vector convention.

```Python
from shenfun import *
import sympy as sp
from IPython.display import Math, Latex, display
from scipy.sparse.linalg import eigs
config['basisvectors'] = 'covariant'

phi, t = psi = sp.symbols('x,y', real=True)

RR = sp.Rational(132, 20)/sp.pi # Same as Kalvoda et al
#RR = 2
```

--------------------------------

### Shenfun FEM Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/shenfun.do.txt

Initializes the Shenfun framework by creating a local mesh and defining the trial and test functions for the finite element method. This prepares the system for assembling the finite element matrices.

```python
from shenfun import TrialFunction, TestFunction, inner, div, grad, Solver

X = T.local_mesh(True) # Assuming T is a pre-defined Shenfun TensorSpace or similar
u = TrialFunction(T)
v = TestFunction(T)
```

--------------------------------

### Install Shenfun with Conda

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Installs Shenfun and its dependencies using conda from the conda-forge channel. It's recommended to install into a fresh environment.

```shell
conda install -c conda-forge shenfun
```

```shell
conda create --name shenfun -c conda-forge shenfun
conda activate shenfun
```

```shell
conda create --name shenfun -c conda-forge shenfun mpich
```

--------------------------------

### Install Matplotlib

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Instructions for installing the Matplotlib library using conda. Matplotlib is an optional dependency for plotting functionalities.

```bash
conda install matplotlib
```

--------------------------------

### Build HTML Documentation (Optimized)

Source: https://github.com/spectraldns/shenfun/blob/master/docs/README.md

A faster way to build HTML documentation if the demo files have not been modified. It skips the regeneration of demo files.

```bash
make ohtml
```

--------------------------------

### Tensor Product Spaces for 2D Poisson Equation

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/shenfun.do.txt

Demonstrates the setup for solving the 2D Poisson equation using tensor product spaces. It shows how to combine different bases (Chebyshev and Fourier) and configure the `TensorProductSpace` for distributed computation. The example highlights the use of `Function` for distributed arrays and the flexibility in defining boundary conditions.

```python
from shenfun.chebyshev.bases import ShenDirichletBasis
from shenfun.fourier.bases import FourierBasis
import numpy as np
from shenfun import Function, TensorProductSpace
from mpi4py import MPI

comm = MPI.COMM_WORLD
N = (32, 33)

K0 = ShenDirichletFunctionSpace(N[0])
K1 = FourierFunctionSpace(N[1], dtype=np.float)
T = TensorProductSpace(comm, (K0, K1))

# Alternatively, switch order for periodic in first direction instead
# T = TensorProductSpace(comm, (K1, K0), axes=(1, 0))
```

--------------------------------

### Build Shenfun Locally

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Installs Shenfun from a local directory after cloning or forking the repository. This is often done after installing dependencies via conda.

```shell
pip install .
```

```shell
python setup.py build_ext -i
```

--------------------------------

### Shenfun Project Link

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap000.html

A simple HTML link to the main page of the Shenfun project, which is hosted on GitHub.

```html
<a href="shenfun_bootstrap.html">Shenfun - automating the spectral Galerkin method</a>
```

--------------------------------

### Install Parallel HDF5 and NetCDF4

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Installs HDF5 and NetCDF4 with MPI support using conda. This is necessary for parallel data storage and retrieval.

```shell
conda install -c conda-forge h5py=*=mpi* netcdf4=*=mpi*
```

--------------------------------

### TensorProductSpace Initialization

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/_sources/shenfun.rst.txt

Demonstrates the creation of a TensorProductSpace using ShenDirichletBasis and FourierBasis. It also shows how to switch the order of bases for different distribution strategies.

```python
from shenfun.chebyshev.bases import ShenDirichletBasis
from shenfun.fourier.bases import FourierBasis
import numpy as np
from shenfun import Function, TensorProductSpace
from mpi4py import MPI

comm = MPI.COMM_WORLD
N = (32, 33)

K0 = ShenDirichletBasis(N[0])
K1 = FourierBasis(N[1], dtype=np.float)
T = TensorProductSpace(comm, (K0, K1))

# Alternatively, switch order for periodic in first direction instead
# T = TensorProductSpace(comm, (K1, K0), axes=(1, 0))
```

--------------------------------

### Shenfun Table of Contents Navigation

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap000.html

HTML structure for the table of contents of the Shenfun documentation, providing links to various sections including Introduction, Spectral Galerkin Method, Shenfun classes, Poisson equation, Tensor product spaces, and more.

```html
<ul>
    <li><a href="#">Contents</a></li>
    <li><strong>Table of contents</strong></li>
    <li><strong>Introduction</strong></li>
    <li><strong>Spectral Galerkin Method</strong></li>
    <li><strong>Shenfun</strong></li>
    <li>Classes for basis functions</li>
    <li>Classes for matrices</li>
    <li>Variational forms in 1D</li>
    <li>Poisson equation implemented in 1D</li>
    <li>Periodic boundary conditions</li>
    <li>Dirichlet boundary conditions</li>
    <li><strong>Tensor product spaces</strong></li>
    <li><strong>Other functionality of `shenfun`</strong></li>
    <li><strong>Ginzburg-Landau equation</strong></li>
    <li><strong>Conclusions</strong></li>
    <li><strong>Acknowledgements</strong></li>
    <li><strong>Bibliography</strong></li>
</ul>
```

--------------------------------

### Build HTML Documentation

Source: https://github.com/spectraldns/shenfun/blob/master/docs/README.md

Generates HTML documentation from docstrings and rst-files. This command also regenerates demo files written in doconce.

```bash
make html
```

--------------------------------

### Run Tests with Pytest

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/installation.rst

Command to execute the test suite for Shenfun using pytest. Ensure pytest is installed in the correct conda environment.

```bash
python -m pytest tests/
```

--------------------------------

### Shenfun Preprocessor Import

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/FastTransforms/fasttransforms.do.txt

Imports the preprocessor module, likely for setting up spectral computations.

```python
#include "../preprocesser.py"
```

--------------------------------

### Tensor Product Space and Function Initialization

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootswatch_cyborg006.html

Sets up the tensor product space with and without padding for de-aliasing, and declares trial, test, and solution functions. It also initializes the solution from a given input.

```Python
# Size of discretization
N = (201, 201)

# Create tensor product space
K0 = C2CBasis(N[0], domain=(-50., 50.))
K1 = C2CBasis(N[1], domain=(-50., 50.))
T = TensorProductSpace(comm, (K0, K1))

Kp0 = C2CBasis(N[0], domain=(-50., 50.), padding_factor=1.5)
Kp1 = C2CBasis(N[1], domain=(-50., 50.), padding_factor=1.5)
Tp = TensorProductSpace(comm, (Kp0, Kp1))

u = TrialFunction(T)
v = TestFunction(T)
X = T.local_mesh(True)
U = Function(T, False)         # Solution
U_hat = Function(T)            # Solution spectral space
Up = Function(Tp, False)       # Padded solution for nonlinear term
dU_hat = Function(T)           # right hand side
#initialize
U[:] = ul(*X)
U_hat = T.forward(U, U_hat)
```

--------------------------------

### Shenfun Surface Integral Jacobian (Example 2)

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/Integration/surfaceintegration.do.txt

Prints the Jacobian determinant for the surface integral calculation in Example 2, which represents the surface area element for the given parametrization.

```python
print(T.coors.sg)
```

--------------------------------

### Shenfun Initialization

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/FastTransforms/fasttransforms.do.txt

Imports necessary libraries from Shenfun and mpi4py-fft for performing spectral computations and projections.

```python
from shenfun import *
from mpi4py_fft import fftw
```

--------------------------------

### RK4 Integrator Setup and Solve

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/KleinGordon/kleingordon.ipynb

Demonstrates the setup and execution of a Runge-Kutta 4th order (RK4) integrator for a simulation. It initializes the integrator with specific parameters and then solves the system over a defined time interval.

```python
par = {'Compute_energy': 10,
       'plot_tstep': 10,
       'end_time': 1}
dt = 0.005
integrator = RK4(TT, N=NonlinearRHS, update=update, energy=[""], **par)
integrator.setup(dt)
fu_hat = integrator.solve(fu, fu_hat, dt, (0, par['end_time']))
```

--------------------------------

### Table of Contents Navigation

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap001.html

Provides a structured list of links for navigating through the Shenfun documentation, including sections like Introduction, Spectral Galerkin Method, and specific equation implementations.

```html
*   [Contents](#)
    *   [**Table of contents**](._shenfun_bootstrap000.html#table_of_contents)
    *   [**Introduction**](#sec:introduction)
    *   [**Spectral Galerkin Method**](._shenfun_bootstrap002.html#sec:preliminaries)
    *   [**Shenfun**](._shenfun_bootstrap003.html#sec:shenfun)
    *      [Classes for basis functions](._shenfun_bootstrap003.html#___sec3)
    *      [Classes for matrices](._shenfun_bootstrap003.html#sec:matrices)
    *      [Variational forms in 1D](._shenfun_bootstrap003.html#___sec5)
    *      [Poisson equation implemented in 1D](._shenfun_bootstrap003.html#___sec6)
    *         [Periodic boundary conditions](._shenfun_bootstrap003.html#sec:fourierpoisson)
    *         [Dirichlet boundary conditions](._shenfun_bootstrap003.html#sec:dirichletpoisson)
    *   [**Tensor product spaces**](._shenfun_bootstrap004.html#sec:tensorproductspaces)
    *   [**Other functionality of `shenfun`**](._shenfun_bootstrap005.html#sec:extended)
    *   [**Ginzburg-Landau equation**](._shenfun_bootstrap006.html#sec:ginzburg)
    *   [**Conclusions**](._shenfun_bootstrap007.html#___sec12)
    *   [**Acknowledgements**](._shenfun_bootstrap008.html#___sec13)
    *   [**Bibliography**](._shenfun_bootstrap009.html#___sec14)
```

--------------------------------

### Shenfun Python Usage Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/kleingordon.ipynb

Provides a conceptual example of how Shenfun can be used to implement the inner product (nabla u, nabla v), referencing a specific section for a Runge-Kutta integrator implementation.

```Python
# Conceptual example of using Shenfun for (nabla u, nabla v)
# Refer to Sec. Runge-Kutta integrator for implementation details
# from shenfun import ...
# ... implementation details ...
```

--------------------------------

### Tensor Product Space Setup and Biharmonic Solver

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap004.html

Demonstrates setting up a tensor product space with different bases (ShenBiharmonicBasis, C2CBasis, R2CBasis) and solving a biharmonic equation using the Biharmonic solver. It includes defining trial and test functions, computing inner products, and performing the backward transform to real space.

```python
from shenfun.chebyshev.bases import ShenBiharmonicBasis, Basis
from shenfun.fourier.bases import R2CBasis, C2CBasis
from shenfun.chebyshev.la import Biharmonic as Solver
from shenfun import inner , div , grad , TestFunction, TrialFunction, project, Dx
from shenfun import Function, TensorProductSpace
from mpi4py import MPI
import numpy as np

comm = MPI.COMM_WORLD

N = (32, 33, 34)
K0 = ShenBiharmonicBasis(N[0])
K1 = C2CBasis(N[1])
K2 = R2CBasis(N[2])
W = TensorProductSpace(comm, (K0, K1, K2))
u = TrialFunction(W)
v = TestFunction(W)
matrices = inner(v, div(grad(div(grad(u)))))
fj = Function(W, False)
fj[:] = np.random.random(fj.shape)

f_hat = inner(v, fj) # Some right hand side
B = Solver(**matrices)

# Solve and transform to real space
u_hat = Function(W) # Solution spectral space
u_hat = B(u_hat , f_hat) # Solve
u = Function(W, False)
u = W.backward(u_hat, u)
```

--------------------------------

### SparseMatrix Class Methods

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/_sources/shenfun.rst.txt

Illustrates the core methods of the SparseMatrix class, including initialization, conversion to Scipy sparse format, matrix-vector product, and solving linear systems.

```python
class SparseMatrix(dict):
    def __init__(self, d, shape):
        dict.__init__(self, d)
        self.shape = shape
        
    def diags(self, format='dia'):
        """Return Scipy sparse matrix"""

    def matvec(self, u, x, format='dia', axis=0):
        """Return Matrix vector product self*u in x"""
        
    def solve(self, b, u=None, axis=0):
        """Return solution u to self*u = b"""
```

--------------------------------

### Shenfun Python Usage Example

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/KleinGordon/kleingordon.ipynb

Provides a conceptual example of how Shenfun can be used to implement the inner product (nabla u, nabla v), referencing a specific section for a Runge-Kutta integrator implementation.

```Python
# Conceptual example of using Shenfun for (nabla u, nabla v)
# Refer to Sec. Runge-Kutta integrator for implementation details
# from shenfun import ...
# ... implementation details ...
```

--------------------------------

### Basis Functions for Spectral Methods

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/introduction.rst

Examples of basis functions used in spectral methods for one spatial dimension, including Chebyshev and Legendre polynomials, and trigonometric functions. Some examples satisfy homogeneous Dirichlet and Neumann boundary conditions.

```mathematica
\phi_k(x) = T_k(x)
```

```mathematica
\phi_k(x) = T_k(x) - T_{k+2}(x)
```

```mathematica
\phi_k(x) = L_k(x)
```

```mathematica
\phi_k(x) = L_k(x) - L_{k+2}(x)
```

```mathematica
\phi_k(x) = \exp(\imath k x)
```

```mathematica
\phi_k = T_k-\left(\frac{k}{k+2}\right)^2T_{k+2}
```

```mathematica
\phi_k = L_k-\frac{k(k+1)}{(k+2)(k+3)}L_{k+2}
```

--------------------------------

### Shenfun Package Basics

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/mekit17/pub/._shenfun_bootstrap001.html

Describes the fundamental aspects of the `shenfun` package, including implementations for simple 1D Poisson and biharmonic problems. This provides a starting point for understanding how Shenfun is used for basic PDE solving.

```APIDOC
Shenfun Package Basics:
  Description: Covers the basics of the `shenfun` package.
  Implementations: Includes examples for 1D Poisson and biharmonic problems.
  Related Sections: 'Spectral Galerkin Method', 'Tensor product spaces'.
```

--------------------------------

### Shenfun Curvilinear Coordinates Example (Sphere)

Source: https://github.com/spectraldns/shenfun/blob/master/README.rst

This example showcases Shenfun's ability to solve PDEs on curvilinear grids, specifically on a sphere using spherical coordinates. It illustrates how Shenfun automatically handles differential operators in non-Cartesian systems.

```Python
from shenfun import * 

# Define spherical coordinates and grid
# ... (code for spherical coordinate system) ...

# Define PDE in spherical coordinates
# ... (code for PDE formulation) ...

# Solve and visualize
# ... (code for solving and plotting results) ...
```

--------------------------------

### Rayleigh Benard Convection Simulation Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/demos/RayleighBenard/rayleighbenard.ipynb

This snippet outlines the setup for simulating Rayleigh-Benard convection using the shenfun Python module. It involves solving the Navier-Stokes and temperature equations in a 2D channel with specific boundary conditions. The simulation is designed to run efficiently with MPI.

```Python
import shenfun

# ... (rest of the simulation setup and solver logic)
```

--------------------------------

### Shenfun and Sympy Implementation Setup

Source: https://github.com/spectraldns/shenfun/blob/master/docs/source/polarhelmholtz.ipynb

This Python code snippet demonstrates the initial setup for implementing spectral methods using the Shenfun library. It imports necessary functionalities from Shenfun and Sympy, and configures Shenfun to use covariant basis vectors, which is relevant for handling polar coordinates.

```python
from shenfun import *
import sympy as sp
config['basisvectors'] = 'covariant'
```
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